Young's modulus is a parameter of given material describing relation between force and deformation.
The formula is
E = σ/ε, where
E is Young's modulus,
σ = F/A is the tensile stress, or force
F over the cross-sectional area
A (e.g. string cross-section), and
ε = (l - L0)/L0 is relative change of length. The units of
E are Pa (pascals); one could visualize it as pressure occurring in the material in response to deformation. https://en.wikipedia.org/wiki/Young%27s_modulus https://en.wikipedia.org/wiki/Stress_(mechanics) https://en.wikipedia.org/wiki/Deformation_(physics)
Young's modulus of the string material is one of the parameters determining how strings are susceptible to intentional or unintentional bending, and how the pitch may vary due to string elongation due to vibration. E.g. listen to pitch change of long notes played on steel strings in the intro to this song:
Young's modulus also determines the relative longitudinal and transverse force exerted by the string on the bridge due to vibrations [Fletcher, Rossing, The Physics of Musical Instruments]. I presume this affects differences between classical (Nylon strings, Young's modulus of 2–4 GPa) and acoustic (steel strings, Young's modulus ~40 times larger) guitars, I however can't provide any details.
Young's modulus is an important parameter, but not the only one to consider. E.g. Nylon strings have much lower value than steel strings, but they have also lower yield and ultimate strength (45, 70 MPa) in comparison to e.g. piano wire (2000–3000 MPa https://en.wikipedia.org/wiki/Yield_(engineering)). As a result Nylon strings are typically made thicker and are tuned at lower tension (so they don't break), which counteracts some of the effect of large difference in Young's modulus. Thicker nylon strings also absorb energy of higher harmonics, resulting in more mellow sound.
Young's modulus determines speed of sound in the material. E.g. for an elongated rod (like a string) it is
c² = E/ρ where
c is the speed of sound and
ρ is the density. (https://en.wikipedia.org/wiki/Speed_of_sound#One-dimensional_solids) I presume this must be an important factor when choosing materials to make an instrument, especially larger parts with resonances significant for the instrument sound.